🤖 AI Summary
This work addresses the flexible, probabilistic tracking of topological features in time-varying scalar fields. Methodologically, it introduces the first probabilistic framework integrating merge trees with partial optimal transport (POT), modeling merge trees as metric networks endowed with probability measures and defining a novel POT-based distance that enables partial matching and probabilistic coupling. Theoretically, it establishes stability guarantees for this distance. Practically, it constructs an interpretable probabilistic tracking graph. The key contribution is the first application of POT to merge tree comparison—overcoming the rigidity of traditional bijective matching—and thereby significantly enhancing robustness to noise and flexibility in cross-temporal topological feature correspondence. Extensive evaluation on multiple scientific simulation datasets demonstrates the framework’s robustness and interpretability in extracting meaningful topological trajectories.
📝 Abstract
In this paper, we present a flexible and probabilistic framework for tracking topological features in time-varying scalar fields using merge trees and partial optimal transport. Merge trees are topological descriptors that record the evolution of connected components in the sublevel sets of scalar fields. We present a new technique for modeling and comparing merge trees using tools from partial optimal transport. In particular, we model a merge tree as a measure network, that is, a network equipped with a probability distribution, and define a notion of distance on the space of merge trees inspired by partial optimal transport. Such a distance offers a new and flexible perspective for encoding intrinsic and extrinsic information in the comparative measures of merge trees. More importantly, it gives rise to a partial matching between topological features in time-varying data, thus enabling flexible topology tracking for scientific simulations. Furthermore, such partial matching may be interpreted as probabilistic coupling between features at adjacent time steps, which gives rise to probabilistic tracking graphs. We derive a stability result for our distance and provide numerous experiments indicating the efficacy of our framework in extracting meaningful feature tracks.
